Standing Waves: Meaning, Important Terms, Equation

Standing Waves: We all know that light and sound is a wave? Light can undergo reflection due to a mirror and refraction due to a glass slab. A sound wave can also undergo reflection (also known as echo) and refraction but are the phenomenon reflection and refraction limited to sound and light waves or does it happen for other waves? What is a standing wave? How can it be represented? What are its properties? How can it be produced? Does the transfer of energy or momentum take place in a standing wave? Let us read this article to explore more about standing waves.

Waves

Waves are disturbances that propagate from one place to another and transfer energy and momentum. Example: Light waves are generated from the sun, and they get transferred to the earth. Similarly, the sound wave also propagates from one place to the other, transferring energy and momentum. Waves that are formed o the surface of the water transfers energy and momentum wherever they propagate.

If the equation of the disturbance is a simple harmonic motion, then the wave obtained due to that disturbance is a harmonic wave.

Important Terms Related to Standing Waves

There are few important terms related to the wave. Let us understand them first.

Fig: A Travelling Wave

Harmonic wave: Waves that have the sinusoidal equation are known as harmonic waves.

The properties of harmonic waves are as follows:

1. The disturbance performs the simple harmonic motion.
2. Amplitude remains constant.
3. Frequency remains constant.
4. It is a doubly periodic phenomenon. That is, the wave is periodic in time and periodic in space as well. In simple words, if we chose any position, then the wave will be periodic with time at that position and repeat itself after a fixed interval of time, and if weze time and move in space, then the wave will be periodic in space and repeat itself after a fixed distance.
   Time-period: It is the time interval after which the wave repeats itself.
   Phase: It is the state of the particle when the wave is passing through it.
   Wavelength: It is the distance after which the wave repeats itself.
   The frequency, wavelength, and the wave velocity are related as,
   

\(u\, = \,\lambda f\)

Frequency is the inverse of the time period.

Equation of Wave

Hamonic wave is produced by a disturbance that is under simple harmonic motion.

The equation of the disturbance is that of an SHM, i.e.,

\(y\, = \,f(t)\)

Let the velocity of the wave in the medium be u; then the time take for this disturbance to travel a distance \(x\) is, \(t = \,\frac{x}{u}\) Therefore a particle at any distance \(x\) and at any time instance \(t\) will have the same state as that of the initial disturbance a \(t = \,t – \frac{x}{u}\) therefore equation of the wave traveling in the positive x-axis is given as,

\(y\, = \,f\left( {t – \frac{x}{u}} \right)\)

\(y\, = \,A\,\sin \,\left( {\omega \,\left( {t – \frac{x}{u}} \right)} \right)\)

If the wave is traveling in the negative x-axis, then the equation of the wave is,

\(y\, = \,f\left( {t + \frac{x}{u}} \right)\)

\(y = A\sin \left( {\omega \left( {t + \frac{x}{u}} \right)} \right)\)

Equation of the wave can also be represented as,

1. In terms of time-period and wavelength.

\(y = A\,\sin \,\left( {\frac{t}{T} – \frac{x}{\lambda }} \right)\)

2. In terms of angular frequency and wave number,

\(y = A\,\sin \,(\omega t\, – kx)\)

The differential equation of wave is,

\(\frac{{{\partial ^2}y}}{{\partial {r^2}}} = {u^2}\frac{{{\partial ^2}y}}{{\partial {x^2}}}\)

Where,

\(u\) is the wave velocity.

Reflection and Refraction of Wave

Fig: Reflection and Refraction of a Wave

Waves, when incident on the boundary of a medium, can undergo reflection, refraction, or both. In the case of reflection, the resultant wave in the incident medium is the superposition of the resultant and reflected wave.

Two mediums are considered to be different if they have different wave velocities for the given wave.

Let the incident wave be,

\({y_i} = {A_i}\sin \,(\omega t\, – \,kx)\)

Let the velocity of the wave in the first medium be \({u_1}\)

And the velocity of the wave in the second medium be \({u_2}\)

If the wave is reflected from the boundary of a rarer medium, then the wave gets reflected in the same phase.

Fig: Reflection and Refraction of a Wave

Fig: Reflection and Refraction of a Wave

A rarer medium has a greater wave velocity than a denser medium.

Fig: \({u_1}\, > \,{u_2}\)

What are Standing Waves?

When two waves coming from opposite directions superimpose, then the resultant wave formed is called the standing wave. Since two identical waves move in the opposite direction, there is no net flow of energy or momentum.

The position of any medium particle is given by the sum of the displacement due to both the waves. Standing waves contains some points known as nodes and antinodes.

Nodes are the point where the destructive interference occurs, and that point has the least amplitude.

Antinode is formed by constructive interference of the two waves. Antinode has the maximum amplitude of the oscillation.

Formation of Standing Waves

Fig: A Standing Wave

In the case of the standing wave, all the particles of the medium perform Simple Harmonic Motion with different amplitudes ranging from zero at the nodes to a maximum at antinodes.

Let the equation of the light wave be,

\({y_1}(x,\,t)\, = \,A\,\sin \,(\omega t\, – \,kx)\, = \,A\,\sin \,\left( {2\pi ft – \frac{{2\pi }}{\lambda }} \right)\)

Where,

\({y_1}\) is the amplitude of the wave.

\(\omega \) is the angular frequency of the wave.

\(f\) is the frequency of the wave.

\(t\) is the time.

Similarly, the equation of the sound wave from the second source is,

\({y_1}(x,\,t)\, = \,A\,\sin \,(\omega t\, – \,kx)\, = \,A\,\sin \,\left( {2\pi ft – \frac{{2\pi }}{\lambda }x} \right)\)

\({y_2}\) is the amplitude of the displacement of the medium particle located at a point.

\(\omega \) is the angular frequency of the wave.

\(f\) is the frequency of the wave.

When the waves overlap, the resultant wave is given by,

\({y_{net}}(x,\,t)\, = \,{y_1}(x,\,t) + {y_2}(x,\,t)\)

Putting in the values we get,

\({y_{net}}\, = \,A\,\sin \,(2\pi ft – \frac{{2\pi }}{\lambda }x) + A\sin \left( {2\pi ft + \frac{{2\pi }}{\lambda }x} \right)\)

Using the trigonometric property,

\(\sin (A) + \sin (B) = 2\cos \left( {\frac{{A – B}}{2}} \right)\sin \left( {\frac{{A + B}}{2}} \right)\)

\( \Rightarrow {y_{net}} = 2A\cos (\omega t)\sin (2\pi ft – kx)\)

On comparing with the equation of wave,

\( \Rightarrow {y_{net}} = 2A\cos (\omega t)\sin (2\pi ft – kx)\)

The amplitude of the resultant wave is given by,

\(A(x) = 2A\sin \left( {\frac{{2\pi }}{\lambda }x} \right)\)

Thus, the amplitude of the resultant will be a function of the position of the point.

Fig: Standing wave

For the maximum amplitude,

\({A_{net}} = 2A\sin \left( {\frac{{2\pi }}{\lambda }x} \right)\) It should be maximum.

\(\sin \,\left( {\frac{{2\pi }}{\lambda }\,x} \right)\) Should be; \( \pm 1\)

\(\frac{{2\pi }}{\lambda }x = \left( {2n + 1} \right)\frac{\pi }{2}\)

\( \Rightarrow x = \frac{{(2n + 1)}}{2}\lambda \)

For zero amplitude,

\({A_{net}} = 2A\sin \left( {\frac{{2\pi }}{\lambda }x} \right)\) It should be \(0\)

\(\frac{{2\pi }}{\lambda }({x_2} – {x_1}) = n\pi \)

\( \Rightarrow x = \frac{n}{2}\lambda \)

Organ Pipe

Closed organ pipe: For a closed organ pipe, the boundary is rigid, and thus, the displacement wave gets reflected in the opposite phase. That is, the phase difference between the reflected and incident wave becomes π, but the pressure wave is reflected in the same phase.

For a closed organ pipe, the minimum length of the pipe required for the formation of the standing wave is \(\frac{\lambda }{4}\)

Fig: Closed Organ Pipe

Fundamental frequency for a closed organ pipe,

\({f_1} = \frac{y}{{4l}}\)

Where,

\(l\) is the length of the open organ pipe.

\(v\) is the velocity of the sound.

\({f_1}\) is the fundamental frequency

Open organ pipe: For an open organ pipe, the boundary isn’t rigid, and the displacement wave will get reflected in the same phase, whereas the pressure wave will be reflected in the opposite phase.

For an open organ pipe, the minimum length of the pipe required for the formation of the standing wave is \(\frac{\lambda }{2}\)

Fig: Open Organ Pipe

Fundamental frequency for a closed organ pipe,

\({f_1} = \frac{y}{{2l}}\)

Where,

\(l\) is the length of the open organ pipe.

\(v\) is the velocity of the sound.

 \({f_1}\) is the fundamental frequency

Solved Examples on Standing Waves

Q.1. Derive the expression for second overtone or third harmonic in the case of an open organ pipe.
Ans: Let the length of the open organ pipe be, \(l\)
Let the velocity of the sound wave is \(v\)
Let the third harmonic or second overtone be \(v\)
For the second overtone, we will have one complete wavelength and two one-fourth of the wavelength.

Fig: Second Overtone

\(l = \lambda  + \frac{\lambda }{4} + \frac{\lambda }{4} = \frac{{3\lambda }}{2}\)
\(f = \frac{{3\lambda }}{{2l}}\)
Where,
\(l\) is the length of the open organ pipe.
\(v\) is the velocity of the sound.
\(f\) is the fundamental frequency

Summary

Waves are disturbances that propagate through space. Waves can undergo reflection and refraction at the boundary of the medium. If the wave is reflected from the boundary of a rarer medium, then the wave gets reflected in the same phase. If the wave is reflected from the boundary of a denser medium, then the wave gets reflected in the opposite phase; that is, the phase difference between the reflected and incident wave becomes \(\pi\). A rarer medium has a greater wave velocity than a denser medium. Standing waves are formed due to the interference of coherent waves. For sound waves, the standing waves are formed in an open and closed organ pipe.

Frequently Asked Questions (FAQs)

Q.1. What is a wave?
Ans: Waves are a phenomenon that propagates from one place to another and transfers energy and momentum. Example: Light waves are generated from the sun, and they get transferred to the earth.

Q.2. How are standing waves formed?
Ans: Standing waves are formed by the interference of coherent waves. On a string, it is formed y superposition of the incident and the reflected wave.

Q.3. Is beats a standing wave?
Ans: No, beats are not standing waves. Beats are formed by the interference of incoherent waves. 

Q.4. What are nodes and antinodes?
Ans: Nodes are the point in a standing wave whose amplitude is zero. That is, they are always at rest. Antinodes are points with the greatest amplitude.

Q.5. Do energy and momentum are transferred in a standing wave?
Ans: No, energy and momentum are not transferred in a standing wave.

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